Multiple periodic solutions of a ratio-dependent predator–prey model

Abstract

A delayed ratio-dependent predator–prey model with non-monotone functional response is investigated in this paper. Some new and interesting sufficient conditions are obtained for the global existence of multiple positive periodic solutions of the ratio-dependent model. Our method is based on Mawhin’s coincidence degree and some estimation techniques for the a priori bounds of unknown solutions to the equation Lx = λNx. An example is represented to illustrate the feasibility of our main result.

Introduction

The dynamic relationship between predator and its prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. The traditional predator–prey models has been studied extensively (e.g. see [1], [2], [3], [4], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37] and references cited therein), but they are questioned by several biologists. Thus, the Lotka–Volterra type predator–prey model with Holling type II functional response has proposed and well studied. The model takes the form of

$$\left\{\begin{array}{c}\frac{\mathrm{d}x}{\mathrm{d}t}=x[a-\mathit{bx}]-\mathit{cy}\frac{x}{m+x}\text{,}\hfill \\ \frac{\mathrm{d}y}{\mathrm{d}t}=y\left[-d+\frac{\mathit{fx}}{m+x}\right]\text{,}\hfill \end{array}\right.$$

where x(t) and y(t) stand for the densities of the prey and the predator, respectively, a, c, d and f are the prey intrinsic growth rate, capture rate, death rate of the predator, and conversion rate respectively, ${\displaystyle \frac{a}{b}}$ gives the carrying capacity of the prey, and m is the half saturation constant. Note that here the functional response $g(x)={\displaystyle \frac{\mathit{cx}}{m+x}}$ is prey-dependent only.

However, models with such prey-dependent-only response function have been facing challenges from biology and physiology communities (see e.g. [5], [6], [7], [8], [9], [10], [11], [12]). Based on growing explicit biological and physiological evidences, they argued that in many situations, especially when predators have to search for food (and therefore have to share or compete for food), a more suitable general predator–prey theory should be based on the so-called ratio-dependent theory, which can be roughly stated as that the per capita predator growth rate should be a function of the ratio of prey to predator abundance, and so should be the so-called predator functional responses. This has been strongly supported by numerous field and laboratory experiments and observations. For this reason, Arditi and Ginzburg [6] first proposed following ratio-dependent predator–prey model:

$$\left\{\begin{array}{c}\frac{\mathrm{d}x}{\mathrm{d}t}=x[a-\mathit{bx}]-\frac{\mathit{cxy}}{\mathit{my}+x}\text{,}\hfill \\ \frac{\mathrm{d}y}{\mathrm{d}t}=y\left[-d+\frac{\mathit{fx}}{\mathit{my}+x}\right].\hfill \end{array}\right.$$

Here, the functional response is the ratio-dependent response $g(x\text{,}y)={\displaystyle \frac{\mathit{cx}/y}{m+x/y}}$. For detailed justifications of (2) and its merits versus (1), see Arditi and Ginzburg [6], [7], [8], [9], Berryman [11], and Lundberg and Fryxell [12]. For the mathematical aspect, Beretta and Kuang [13], Kuang and Beretta [14], Hsu et al. [15] and Jost et al. [16] have studied and given a nice systematic work on the global qualitative analysis of the ratio-dependent system (2).

As we know, the variation of the environment plays an important role in many biological and ecological dynamical systems. In particular, the effects of a periodically varying environment are important for evolutionary theory as the selective forces on system in a fluctuating environment differ from those in a stable environment. To incorporate the varying property of the parameters into the model, Fan et al. [17] considered the following non-autonomous ratio-dependent model:

$$\left\{\begin{array}{c}\frac{\mathrm{d}x}{\mathrm{d}t}=x[a(t)-b(t)x]-\frac{c(t)\mathit{xy}}{m(t)y+x}\text{,}\hfill \\ \frac{\mathrm{d}y}{\mathrm{d}t}=y\left[-d(t)+\frac{f(t)x}{m(t)y+x}\right].\hfill \end{array}\right.$$

They obtained a set of criteria which guarantee the permanence, non-persistence, global asymptotic stability, the existence and uniqueness of periodic and almost periodic solution of system (3).

Note that the ratio-dependent response $g(x\text{,}y)={\displaystyle \frac{\mathit{cx}/y}{m+x/y}}$ in system (2), (3) is the monotone functional response. In microbial dynamics or chemical kinetics, the functional response describes the uptake of substrate by the microorganisms. In general, the response function g(x) or g(x, y) is monotone. However, there are experiments that indicate that non-monotonic functional responses occur at the microbial level: when the nutrient concentration reaches a high level an inhibitory effect on the specific growth rate may occur. This is often seen when microorganisms are used for waste decomposition or for water purification (see [18]). The so-called Monod–Haldane function $g(x)={\displaystyle \frac{\mathit{cx}}{m^2+\mathit{bx}+x^2}}$ has been proposed and used to model the inhibitory effect at high concentrations (see [19]). In experiments on the uptake of phenol by pure culture of Pseudomonas putida growing on phenol in continuous culture, Sokol and Howell [20] proposed a simplified Monold–Haldane function of the form $g(x)={\displaystyle \frac{\mathit{cx}}{m^2+x^2}}$ and found that it fits their experimental data significantly better than the Monod–Haldae function. Unlike the monotone response, the non-monotonic response is humped and declines at high prey density.

According to the ratio-dependent theory introduced above, a more general and more realistic non-monotone functional response should be proposed. Therefore, to incorporate the ratio-dependent theory and the inhibitory effect on the specific growth rate into the predator–prey model, we propose the following periodic ratio-dependent model with non-monotone functional response:

$$\left\{\begin{array}{c}\frac{\mathrm{d}x(t)}{\mathrm{d}t}=x(t)\left[a(t)-b(t){\int }_{-\infty }^{t}K(t-s)x(s)\mathrm{d}s-\frac{c(t)y^2(t)}{m^2{y}^2(t)+x^2(t)}\right]\text{,}\hfill \\ \frac{\mathrm{d}y(t)}{\mathrm{d}t}=y(t)\left[-d(t)+\frac{e(t)x(t-\tau (t))y(t-\tau (t))}{m^2{y}^2(t-\tau (t))+x^2(t-\tau (t))}\right]\text{,}\hfill \end{array}\right.$$

where x(t) and y(t) represent predator and prey densities, respectively; $a(t)\text{,}b(t)\text{,}c(t)\text{,}e(t)\text{,}d(t)$ and τ(t) are all positive periodic continuous functions with period ω > 0, m is a positive real constant, K(s) : R₊ ↦ R₊ is a measurable, ω-periodic, normalized function such that ${\int }_0^{+\infty }K(s)\mathrm{d}s=1$; corresponding to a delay kernel or a weighting factor, which says how much emphasis should be given to the size of the prey population at earlier times to determine the present effect on resource availability. In this case, $g(x\text{,}y)={\displaystyle \frac{\mathit{cx}/y}{m^2+(x/y{)}^2}}$.

Let τ = max_{t ∈ [0,ω]}{τ(t)}. From the point of view of mathematical biology, we choose

$$\mathcal{C}=\{\phi =({\phi }_1\text{,}{\phi }_2{)}^{\mathrm{T}}\in \mathcal{C}((-\infty \text{,}0):R_{+}^2):{\phi }_1(0)>0\text{and}{\phi }_2(0)>0\}$$

as the state space for (4), where $R_{+}^2=\{(u\text{,}v)\in R^2:u⩾0\text{,}v⩾0\}$. Then for φ = (φ₁, φ₂)^T and t₀ ∈ R, it is easy to see that system (4) has a unique solution (x(t), y(t))^T = (φ₁(t − t₀), φ₂(t − t₀))^T for t ∈ (−∞, t₀] and (x, y)^T is continuous on R and x(t) > 0, y(t) > 0 for t ⩾ 0.

Recently, the powerful and effective method of coincidence degree has been applied to study the existence of periodic solutions in population models (see e.g. [21], [22], [23], [24], [25], [26], [27], [37], [38]). Motivated by these works, we study system (4) and obtain sufficient conditions for the existence of multiple positive periodic solutions to (4) by applying the method of coincidence degree. It is the first time that multiple periodic solution are obtained for the ratio-dependent model with non-monotone functional response.